一子级可重复使用火箭上升及返回轨迹联合优化

doi:10.7527/S1000-6893.2025.32240

Abstract

Abstract:

The joint optimization of ascent and return trajectories for first-stage reusable rockets involves highly nonlinear multi-stage trajectory planning， posing significant challenges to be solved. To enhance the reliability and optimality of the joint optimization of ascent and return trajectories， this paper proposes a bi-level optimization framework consisting of inner-layer trajectory optimization and outer-layer mission parameter optimization. By synergistically combining trajectory optimization’s constraint-handling capability with surrogate model optimization’s efficiency in black-box function minimization， this framework effectively reduces nonlinearity in trajectory planning and complexities of parametric optimization. For the inner-layer trajectory optimization， we solve the ascent trajectory using an existing convex-optimization-based method. The first-stage return trajectory， which comprises powered deceleration， atmospheric reentry， and powered landing phases， poses greater challenges to be solved. To ensure reliable inner-layer solutions， we develop a convergence-guaranteed trajectory optimization algorithm for first-stage return flight. By analyzing the characteristics of the optimal control profile and parametrizing it， the algorithm designs analytical parameter iteration equations， requiring solving only two parameters to obtain the optimal trajectory， with theoretical convergence guarantees. For the outer-layer mission parameter optimization， we adopt surrogate-model-based optimization algorithm to solve optimal mission parameters， including stage separation states and the mass of each rocket stage， thereby obtaining the optimal trajectory solution that maximizes the payload mass. Simulation results show that， compared with existing methods， the proposed bi-level optimization framework improves payload mass by approximately 25%. Furthermore， the first-stage return trajectory optimization algorithm achieves high computational efficiency， with a runtime of less than 50 ms， demonstrating both reliability and efficiency.

Key words: first-stage reusable rockets, bi-level optimization framework, trajectory optimization, surrogate model, launch capacity

CLC Number:

V475.1

Yan WANG, Qingyun DOU, Guangwei WANG, Yaxuan LI, Xiaoyu HE, Xinfu LIU. Joint optimization for ascent and return trajectories of first-stage reusable rockets[J]. Acta Aeronautica et Astronautica Sinica, 2026, 47(3): 232240.

Figures/Tables 20

Table 1

Mission parameters

符号	含义
$R 1$	一子级质量占火箭总质量的比例
$R 1, r$	一子级返回段预留的燃料占一子级燃料加注质量的比例
$h 1, f$	一子级上升段的终端高度
$γ 1, f$	一子级上升段的终端当地弹道倾角
$ψ 1, f$	一子级上升段的终端当地弹道偏角
$t 2, f$	二子级上升段的终端时间

Table 1

Fig.1

Table 2

Range of initial states and αA， αB

参数	$h 3 (t 3,0)$ /km	$V 3 (t 3,0)$ /（m·s^-1）	$γ 3 (t 3,0)$ /（°）	$ψ 3 (t 3,0)$ /（°）	$α A$ /（°）	$α B$ /（°）
最大值	70	2 500	30	5	10	10
最小值	45	1 800	10	-5	-10	-10

Table 2

Fig.2

Table 3

Rocket’s parameters

模块	参数	数值
全箭	起飞质量 $m 0$ /t	580
全箭	参考面积 $S r e f$ / $m 2$	8.814
一子级	结构系数 $s 1$	0.07
	发动机数量 $N 1$	9
	单台发动机真空推力 $T 1, v a c$ /kN	916.7
	单台发动机真空比冲 $I 1, s p$ /（m·s^-1）	3 046
	单台发动机喷口面积 $A 1, e$ /m²	0.9
二子级	结构系数 $s 2$	0.06
	发动机数量 $N 2$	1
	单台发动机真空推力 $T 2, v a c$ /kN	1 017
	单台发动机真空比冲 $I 2, s p$ /（m·s^-1）	3 413

Table 3

Table 4

Range of mission parameters

参数	$R 1$ /%	$R 1, r$ /%	$h 1, f$ /km	$γ 1, f$ /（°）	$ψ 1, f$ /（°）	$t 2, f$ /s
最大值	85	10	70	30	5	1 800
最小值	65	2	45	10	-5	900

Table 4

Fig.3

Fig.4

Fig.5

Fig.6

Table 5

Comparison results of mission parameters

参数	$R 1$ /%	$R 1, r$ /%	$h 1, f$ /km	$γ 1, f$ /（°）	$ψ 1, f$ /（°）	$t 2, f$ /s
$P *$	76.7	3.91	67.1	10.6	0.58	1 424.5
$P c$	79.6	4.85	48.6	16.1	0.26	1 416.2
$P 0$	79.0	7.00	60.0	20.0	0	1 300

Table 5

Table 6

Comparison results of trajectories

工况

返回段燃料

消耗质量/kg

返回段燃料

剩余质量/kg

有效载荷

质量/kg

工况 1

1.60 × 104

4.37 × 10 - 2

1.81 × 104

工况 2

2.16 × 104

2.98 × 10 - 1

1.77 × 104

工况 3

2.95 × 104

2.55 × 103

1.65 × 104

Table 6

Fig.7

Table 7

Comparison results with existing method

方法

一子级返回段燃料消耗质量/kg

有效载荷

质量/kg

本文方法

1.60 × 104

1.81 × 104

文献［4］

2.31 × 104

1.44 × 104

Table 7

Fig.8

Fig.9

Fig.10

Table 8

Range of initial states

参数	$h 3 (t 3,0)$ /km	$γ 3 (t 3,0)$ /°	$ψ 3 (t 3,0)$ /°	$V 3 (t 3,0)$ /（m·s^-1）	$m 3 (t 3,0)$ /kg
最大值	70	30	5	2 200	60 000
最小值	45	10	-5	1 500	40 000

Table 8

Fig.11

Fig.12

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Joint optimization for ascent and return trajectories of first-stage reusable rockets

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